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克利福德代数

读物

• 维基百科: Spinor
• 维基百科: Spin group
• 维基百科: Spin representation Defines spinors in terms of spin group representations and Clifford algebras.
• Higher-dimensional gamma matrices
• Jose Figueroa-O'Farrill. Majorana Spinors. Used by Chris. Manifest Bott periodicity.
• Fulton and Harris – Representation Theory: A First Course
• Jean Gallier. Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups. 2012. Includes a proof of Bott periodicity.
• The lecture notes on Clifford algebras by Lundholm and Svensson.
• Varadarajan - Supersymmetry for mathematicians

Spinor

• Low dimension spinor representations have 8fold periodicity https://en.m.wikipedia.org/wiki/Spinor

Harvey, F. Reese (1990), "Chapter 2: The Eight Types of Inner Product Spaces", Spinors and calibrations, Academic Press, pp. 19–40, ISBN 0-12-329650-1

MacMahon Master Theorem

• The coefficient of {$x_1^{k_1}\cdots x_m^{k_m}$} in {$\frac{1}{\det (I_m - TA)}$} equals its coefficient in {$\prod_{i=1}^m \bigl(a_{i1}x_1 + \dots + a_{im}x_m \bigl)^{k_i}$}
• My thesis has combinatorial interpretations for the generating function {$\frac{1}{\det (I - A)} = \sum_{n=0}^{\infty}x^nh_n(\xi_1,...,\xi_n)$}
• Julian Schwinger: It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents.
• In the Standard Model, fermions are not their own antiparticles, but in some theories they can be. Among other things, this involves the question of whether the relevant spinor representations of the groups Spin(p,q) are complex, real (‘Majorana spinors’) or quaternionic (‘pseudo-Majorana spinors’). The options are well-understood, and follow a nice pattern depending on the dimension and signature of spacetime modulo 8.

Twistors

https://en.wikipedia.org/wiki/Twistor_theory

• Do twistors relate the threesome and the foursome?
• Is there an eight-dimensional concept that expands on spinors and twistors?